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Chen Ning Yang on the physics of Yang-Mills theory being about the mathematics of connections on fiber bundles:

$[$This$]$ was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the physical world is intimately tied to the deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and the beauty of form?

In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of S. S. Chern in El Cerrito, near Berkeley… I said I found it amazing that gauge theory are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added: “this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.” He immediately protested: “No, no. These concepts were not dreamed up. They were natural and real.

$$\underline{\mathbf{1}} \;\; \colon \;\; \array{ \mathbb{Z}_2/1 &\mapsto& \mathbf{1} \\ \big\downarrow && \big\downarrow{}^{\mathrlap{id}} \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& \mathbf{1} }$$

The injective envelope is

Let the equivariance group be the cyclic group of order 2:

$$G \;\coloneqq\; \mathbb{Z}_2$$

The orbit category looks like this:

(1)$$\mathbb{Z}_2 Orbits \;=\; \left\{ \array{ \mathbb{Z}_2/1 & \overset{ \phantom{AAAAA} }{ \longrightarrow } & \mathbb{Z}_2/\mathbb{Z}_2 \\ Aut = \mathbb{Z}_2 && Aut = 1 } \right\}$$

i.e.:

$$\begin{aligned} G Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; 1 \\ G Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; \ast \\ G Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \emptyset \\ G Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \mathbb{Z}_2 \end{aligned}$$

Consider the topological G-space

$$\prec (X \sslash G) \;\coloneqq\; \prec (S^4 \sslash \mathbb{Z}_2)$$

with $\mathbb{Z}_2$ acting on the 4-sphere $S^4 = S(\mathbb{R}^5)$ by the involution $(x^1, x^2, x^3, x^4, x^5) \mapsto (x^1, x^2, x^3, - x^4, - x^5)$.

Minimal equivariant Sullivan model

Stage 0

$$CE(\mathfrak{l} \prec (X \sslash G) )_0 \;\coloneqq\; \underline{\mathbb{Q}}$$

Stage 1

$$CE(\mathfrak{l} \prec (X \sslash G) )_1 \;\coloneqq\; \underline{\mathbb{Q}}$$

Stage 2

$$CE(\mathfrak{l} \prec (X \sslash G) )_2 \;\coloneqq\; \mathbb{Q} \big[ I_1(\mathbf{1}) \big] \big/ \big( d\, I_1(\mathbf{1}) = 0 \big)$$

Consider the vector G-space given by

$$I_1(\mathbf{1}) \;\; \colon \;\; \array{ \mathbb{Z}_2/1 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & \mathbf{1} \\ \big\downarrow && \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, 0 }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & 0 }$$

$$I_1(\mathbf{1}_{sgn}) \;\; \colon \;\; \array{ \mathbb{Z}_2/1 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1}_{sgn} \Big) & \simeq & \mathbf{1}_{sgn} \\ \big\downarrow && \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& \mathbb{Z}_2 Reps \Big( \underset{ \simeq \, 0 }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 ) \big] } } \,,\, \mathbf{1}_{sgn} \Big) & \simeq & 0 }$$

$$I_{\mathbb{Z}_2}(\mathbf{1}) \;\; \colon \;\; \array{ \mathbb{Z}_2/1 &\mapsto& 1 Reps \Big( \underset{ \simeq \, 0 }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/\mathbb{Z}_2 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & 0 \\ \big\downarrow && \\ \mathbb{Z}_2/\mathbb{Z}_2 &\mapsto& 1 Reps \Big( \underset{ \simeq \, \mathbb{Q} }{ \underbrace{ \mathbb{Q} \big[ \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/\mathbb{Z}_2 ) \big] } } \,,\, \mathbf{1} \Big) & \simeq & \mathbf{1} }$$